On Hardy type inequalities for weighted means

The aim of this paper is to establish weighted Hardy type inequality in a broad family of means. In other words, for a fixed vector of weights $({\lambda }_n{)}_{n=1}^{\mathrm{\infty }}$ and a weighted mean $\mathcal{M}$, we search for the smallest number $C$ such that $$\sum _{n=1}^{\mathrm{\infty }}{\lambda }_n\mathcal{M}((x_1,\dots ,x_n),({\lambda }_1,\dots ,{\lambda }_n))\le C\sum _{n=1}^{\mathrm{\infty }}{\lambda }_n{x}_n\text{ for all admissible }x\mathrm{.}$$ The main results provide a definite answer in the case when $\mathcal{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, if $\mathcal{M}$ is symmetric, Jensen-concave, and the sequence $\left(\frac{{\lambda }_n}{{\lambda }_1+\cdots +{\lambda }_n}\right)$ is nonincreasing. In addition, it is proved that if $\mathcal{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $\lambda$ is the constant vector.